The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 <= 1. What...
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Banach-Tarski Paradox
Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...
Continue reading...Sierpinski-Mazurkiewicz Paradox
If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces...
Continue reading...Face Derivatives and Computer Vision
One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”...
Continue reading...Finding the N-th digit of Pi
Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi =...
Continue reading...Large Counterexample
A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is...
Continue reading...Area of a Circle or Regular Polygon
There’s a nice way to see why the formula for the area of a circle of radius R is: Pi...
Continue reading...Pretzel Unlinking
Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...
Continue reading...Chords of an Ellipse
Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced...
Continue reading...Unbelievable Unlinking
Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber....
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